目前的工作提供了一个明确的答案的问题量化松弛平衡的解决方案的空间均匀玻尔兹曼方程麦克斯韦分子。故事的开始可以追溯到希尔伯特的一项开创性工作，他首先正式提出了碰撞算子线性化的概念，并指出了其本征值对于玻尔兹曼方程的某种渐近行为的重要性。在初始数据非常温和的条件下-接近必要-以及弱的，物理上一致的，角度截止假设，我们的主要结果（定理1.1）包含了第一个精确的陈述，即解与极限麦克斯韦分布之间的总变差距离允许一个形式为Ce ^{Lambda_b t}的上界，Lambda_b是上述特征值中最小的负值，C是一个常数，它只取决于初始数据的一些简单的数值特征（例如矩）。大约50年前，亨利·P·麦基恩（Henry P. McKean）在一篇论文中证实了这种量化的有效性，但是，尽管进行了几次尝试，迄今为止已知的最佳答案在于一个速率可以任意接近λ_b的界限，以常数C的“爆炸”为代价。此外，它的推导受到限制性的假设的初始数据，除了梯度角截止条件。至于我们的结果的证明，我们已经采取了作为参考点之间的类比收敛到平衡的问题和概率论的中心极限定理，突出了McKean。我们的工作实际上是对这种类比的确认，因为我们在这里开发的技术关键依赖于中心极限定理的某些公式。
英文摘要：
The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. The beginning of the story dates back to a pioneering work by Hilbert, who first formalized the concept of linearization of the collision operator and pointed out the importance of its eigenvalues with respect to a certain asymptotic behavior of the Boltzmann equation. Under really mild conditions on initial data - close to being necessary - and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1.1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form C e^{Lambda_b t}, Lambda_b being the least negative of the aforesaid eigenvalues and C a constant which depends only on a few simple numerical characteristics (e.g. moments) of the initial datum. The validity of this quantification was conjectured, about fifty years ago, in a paper by Henry P. McKean but, in spite of several attempts, the best answer known up to now consists in a bound with a rate which can be made arbitrarily close to Lambda_b, to the cost of the "explosion" of the constant C. Moreover, its deduction is subject to restrictive hypotheses on the initial datum, besides the Grad angular cutoff condition. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean. Our work represents in fact a confirmation of this analogy, since the techniques we develop here crucially rely on certain formulations of the central limit theorem.
